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2. (a) Let X be a positive continuous random variable with Vk EN: E X* E R. i. Show that E[X] = S (1- Fx(x))dx. Hint: Write 1 Fx(x) in terms of an integral of the pdf, then switch the order of integration. ii. Find an expression for E[X*] (where k E N) of the form ° (1-Fx(x))h(x)dr for a functionh which you should specify. 1- (1+x)-m if x >0 otherwise (b) Now let Fx(x) = where m e N. i. For which k eN does the expectation E X*] exist? Justify your answer ii. Compute the cdf of the sample minimum X(1) of a random sample of size neN from Fx. For which k eN does E (X(1)* exist?

2. (a) Let X be a positive continuous random variable with Vk EN: E X* E R. i. Show that E[X] = S (1- Fx(x))dx. Hint: Write 1 Fx(x) in terms of an integral of the pdf, then switch the order of integration. ii. Find an expression for E[X*] (where k E N) of the form ° (1-Fx(x))h(x)dr for a functionh which you should specify. 1- (1+x)-m if x >0 otherwise (b) Now let Fx(x) = where m e N. i. For which k eN does the expectation E X*] exist? Justify your answer ii. Compute the cdf of the sample minimum X(1) of a random sample of size neN from Fx. For which k eN does E (X(1)* exist?

A First Course in Probability (10th Edition)
A First Course in Probability (10th Edition)
10th Edition
ISBN:9780134753119
Author:Sheldon Ross
Publisher:Sheldon Ross
Chapter1: Combinatorial Analysis
Section: Chapter Questions
Problem 1.1P: a. How many different 7-place license plates are possible if the first 2 places are for letters and...
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2. (a) Let X be a positive continuous random variable with Vk EN: EX E R. i. Show that E[X] = S (1- Fx(x))dx. Hint: Write 1 Fx(x) in terms of an integral of the pdf, then switch the order of integration. ii. Find an expression for E[X*] (where k E N) of the form (1-Fx(x))h(x)dx for a function h which you should specify. 1- (1+ x)-m if x >0 otherwise (b) Now let Fx(x) = where m e N. i. For which k eN does the expectation E X* exist? Justify your answer ii. Compute the cdf of the sample minimum X(1) of a random sample of size n eN from Fx. For which k EN does E (Xu)*| exist?
Transcribed Image Text:2. (a) Let X be a positive continuous random variable with Vk EN: EX E R. i. Show that E[X] = S (1- Fx(x))dx. Hint: Write 1 Fx(x) in terms of an integral of the pdf, then switch the order of integration. ii. Find an expression for E[X*] (where k E N) of the form (1-Fx(x))h(x)dx for a function h which you should specify. 1- (1+ x)-m if x >0 otherwise (b) Now let Fx(x) = where m e N. i. For which k eN does the expectation E X* exist? Justify your answer ii. Compute the cdf of the sample minimum X(1) of a random sample of size n eN from Fx. For which k EN does E (Xu)*| exist?
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